15 IMP & HIRFL Annual Report · 25 ·

2015 IMP & HIRFL Annual Report · 25 ·

1 - 16 Glueball Quarkonium Mixing in QCD∗

Zou Liping, Zhang Pengming and Xie Jujun

Despite of the fact that the quark model has been very successful in describing hadron spectrum, there are still some exotic states cannot be explained well. Glueball is one of the most interesting issue among the exotic states. The general wisdom is that QCD must have the glueballs made of gluons. And there are many models have been proposed to describe the physics of glueball[1−5]. However, the search for the glueballs has not been so successful both theoretically and experimentally. On experiment side, one reason is the glueballs are expected to have an intrinsic instability which results in the large decay width and short lifetime. The instability comes from the annihilation of gluons and the glueball decay width could be estimated from QCD one-loop effective action[6]. Another reason is that glueballs could mix with quarkoniums, so that glueballs may not exist as mass eigenstates. It implies that we must take care of the possible mixing to identify the glueball. We study the glueball physics based on the theory of Abelian decomposition[7,8]. In this formalism, the gluon filed in QCD can be classified into two types, i.e. the color neutral binding potential(neuron) and colored valence potential(chromon). And glueballs(chromoballs) are constructed by the constituents, chromons. The general picture of the glueballs in our model is very similar to that in the constituent model, so that we can construct color singlet glueballs with gg¯ or ggg . The difference, of course, is that in our case only the six chromons become the constituent gluons because the other two gluons become the binding gluons (the neurons).

Fig. 1 The possible glueball-quarkonium mixing diagrams.

With these, we can discuss the glueball-quarkonium mixing. The possible Feynman diagrams for the mixing is shown in Fig. 1, which tells that the mixing happens not only between the quarkonium and chromoballs but also between the gg and ggg chromonballs. Obviously, the mixing inﬂuences the qq¯ octet-singlet mixing in quark model, so by including the glueball the mixing mass matrix can be generalized as  √  2 2  E + ∆ − ∆ 0   √3 3  M 2 =  2 1 ,  − ∆ E + ∆+3A ν   3 3  0 ν G where A,E,∆ are the parameters of quark octet-singlet mixing, G is the glueball mass parameter and can be expressed with constituent µ mass of chromon, and put G = 4µ2. Diagonalizing the mass matrix one can transform the unphysical states (|8⟩,|1⟩,|G⟩) to the mass eigen states (|m1⟩,|m1⟩,|m1⟩), and obtain the information on the gluon and quark contents of the physical states. With this matrix, one can easily generalize the mass matrix to the 4 × 4 or even 5 × 5 mixing to include more chromon states |G′⟩, which could be gg¯ or ggg.

∗ Foundation item: National Natural Science Foundation of China (11175215, 11447105, 11475227) · 26 · IMP & HIRFL Annual Report 2015

Notice that the mixing mechanism in our model is clearly fixed by Fig. 1. All terms in the matrix have clear physical meaning, and each of them could in principle be calculated from the Feynman diagram within the formalism of ECD. For example, let us discuss the mixing in 0++ channel. Following the PDG suggestion, we choose the parameters 2 2 − 2 ∗ E = a0, a0 = a0(1450), ∆ = 2(K a0), K = K0(1430). ∗ as the input. But notice that here the strange meson of the flavor octet of this channel is K0 (1430), one would expect the mass of the non-strange isotriplet partner to be less than 1 430 MeV. In this case, a0(980) becomes a natural candidate of the isotriplet partner of the flavor octet, and we may choose 2 2 − 2 ∗ E = a0, a0 = a0(980), ∆ = 2(K a0), K = K0(1430).

as the input. However, in this report we discuss only the PDG suggestion, i.e. a0 = a0(1405). As we could see later, with PDG suggestion the prediction of our mixing matrix agrees well with experimental data. As for the mass

eigenstates, we have ﬁve states including f0(500), which we can use as the input. To include all of the ﬁve mass eigenstates we need the 5×5 mixing matrix. For simplicity, we discuss the 3×3 mixing, the numerical analysis is shown in Table 1.

++ Table 1 The numerical analysis of the 3×3 mixing in the 0 channel, with a0(1450), f0(1500), and f0(1710) as the input.

Here the third physical state can be identiﬁed as f0(1370).

µ A ν m3 m1 = f0(1500) m2 = f0(1710) m3 R(m2/m1) R(m3/m1) u+d s G u+d s G u+d s G

0.76 0.27 0.18 1.40 0.07 0.00 0.93 0.73 0.20 0.07 0.19 0.80 0.00 0.05 0.00 0.78 0.23 0.31 1.40 0.26 0.01 0.73 0.59 0.16 0.25 0.15 0.83 0.02 0.14 0.02 0.80 0.18 0.36 1.39 0.44 0.01 0.54 0.45 0.12 0.43 0.11 0.87 0.02 0.59 0.05 0.82 0.14 0.35 1.39 0.62 0.02 0.36 0.30 0.08 0.62 0.09 0.90 0.01 1.26 0.07 0.84 0.09 0.29 1.39 0.79 0.02 0.18 0.15 0.04 0.80 0.05 0.93 0.01 3.26 0.09 0.86 0.04 0.07 1.39 0.96 0.03 0.01 0.01 0.00 0.99 0.03 0.97 0.00 85.71 0.12

The table shows that the third state (with mass around 1 400 MeV) could be identiﬁed to be f0(1370), which becomes predominantly a ss¯ state. But the physical contents of the two other states depend very much on the

value of the chromon mass µ. When µ is around 760 MeV, f0(1500) becomes predominantly a chromoball state and

f0(1710) becomes predominantly the qq¯ state made of u+d. As the chromon mass increases to 860 MeV, f0(1370)

becomes a u + d state and f0(1710) quickly becomes a chromoball state. So when µ ≃ 760 MeV the above result

appears to be in agreement with the suggestion of PDG, which lists f0(1370) and f0(1710) as the qq¯ states. But here the qq¯ state made of s quark becomes lighter than the qq¯ state made of u+d. This, of course, is due to the input

a0(1430).

Notice that the J/Ψ radiative decay branching ratio R[f0(1710)/f0(1500)] ≃ 15.3 (at µ ≃ 856 MeV). This is in

good agreement with the PDG data. This implies that a0(1450), could be the isotriplet partner of these isosinglet states.

Moreover, both recent lattice calculations and phenomenological studies indicate the state f0(1710) to the candidate of lowest lying scalar glueball[9−11]. To favor this argument, it needs us to choose chromon mass around 860 MeV, this is indeed the value we selected above to ﬁt the experimental decay data. References

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